Geodesic
Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 331-348

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An abstract discrete time dynamical system, given by an implicit function of the values of a variable at successive moments of time, is presented. The dynamics of this system is defined ambiguously both in reverse and forward time. An example of a system of such type is described in the works of Bullett, Osbaldestin and Percival [Physica D, 1986, vol. 19, pp. 290–300; Nonlinearity, 1988, vol. 1, pp. 27–50]; it demonstrates some features of the behavior of Hamiltonian systems. The map under study allows a smooth transition from the case of the explicitly defined evolution operator to an implicit one and, further, to the “conservative” limit, corresponding to the symmetric evolution operator satisfying the unitarity condition. Being created on the basis of the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex analytical dynamics to “conservative” phenomena and allows us to identify the relationship between them.
Keywords: Mandelbrot set, Julia set, conservative and quasi-conservative dynamics, multistability
Mots-clés : implicit map. 
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     author = {O. B. Isaeva and M. A. Obychev and D. V. Savin},
     title = {Dynamics of a discrete system with the operator of evolution given by an implicit function: from the {Mandelbrot} map to a unitary map},
     journal = {Russian journal of nonlinear dynamics},
     pages = {331--348},
     publisher = {mathdoc},
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     year = {2017},
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     url = {http://geodesic.mathdoc.fr/item/ND_2017_13_3_a2/}
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O. B. Isaeva; M. A. Obychev; D. V. Savin. Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 331-348. http://geodesic.mathdoc.fr/item/ND_2017_13_3_a2/